The internal common tangents to two circles with centre `C_1` and `C_2` intersect the line joining `C_1` and` C_2` at P and the two direct common tangents intersects the line joining `C_1` and `C_2` at Q. The length `C_1P`, `C_1C_2` are `C_1` Q are in

The internal common tangents to two circles with centre C_(1) and C_(2) intersect the line joining C_(1) and C_(2) at P and the two direct common tangents intersects the line joining C_(1) and C_(2) at Q. The length C_(1)P,C_(1)C_(2) are C_(1) Q are in

The line joining the points (0,3) and (5,-2) becomes tangent to the curve y=c/(x+1) then c=

Two circles with centres at C_(1) , C_(2) and having radii r_(1) , r_(2) will intersect each in two real distinct points if , and only if :

Circles C_1 and C_2 are externally tangent and they are both internally tangent to the circle C_3. The radii of C_1 and C_2 are 4 and 10, respectively and the centres of the three circles are collinear. A chord of C_3 is also a common internal tangent of C_1 and C_2. Given that the length of the chord is (msqrtn)/p where m,n and p are positive integers, m and p are relatively prime and n is not divisible by the square of any prime, find the value of (m + n + p).

In the given figure (not to scale), two circles C_1 and C_2 intersect at S and Q. PQN and RQM are tangents drawn to C_1 and C_2 respectively at Q. MAB and ABN are the chords of the circles C_1 and C_2 . If angle NQR =85^@ , then find angle AQB .